2 The Mathematics of Power

2. 2 The Mathematics of Power 2.1 An Introduction to Weighted Voting 2.2 The Banzhaf Power Index 2.3 Applications of the Banzhaf Power Index 2.4 The Shapley-Shubik Power Index 2.5 Applications of the Shapley-Shubik Power Index

3. The Nassau County (N.Y.) Board of Supervisors Weighted voting in NewYork dates back to the 1800s Weighted voting systems usedin some New York counties were seriously flawed (and unconstitutional) John Banzhaf introduced the Banzhaf Power Index in an article entitledWeighted Voting Doesn’t Work key point wasthat in weighted voting votes do not necessarily imply power

4. The Nassau County (N.Y.) Board of Supervisors 1960s, Nassau County was divided into 6 uneven districts, 4 withhigh populations and 2 rural districts with low populations Table shows names of the 6 districts and their weights based on their respective populations

5. The Nassau County (N.Y.) Board of Supervisors Quota was a simple majority of 58 (out of 115) votes A district’spower should be in proportion to its population, thus ensuring to all citizensthe “equal protection” guaranteed by the Constitution. Banzhaf: Instead of looking at the weights, one should focus on which districts are critical players in the many winning coalitions that can be formed onthe Board.

6. The Nassau County (N.Y.) Board of Supervisors districts–Hempstead #1,Hempstead #2, and Oyster Bay–could ever be critical players, and consequently, the other 3 districts had no power whatsoever. Notice that North Hempstead had no power, never mind its 21votes on the Board! Banzhaf argued that only the 3 largest

7. The Nassau County (N.Y.) Board of Supervisors Banzhaf’s mathematical analysis of the unfair power distribution in the Nassau County Board set the stage for a series of lawsuits against Nassau and otherNew York state counties based on the argument that weighted voting violated the“equal protection” guarantees of the Fourteenth Amendment (which was thevery reason that weighted voting was instituted to begin with).

8. The Nassau County (N.Y.) Board of Supervisors The final resultwas a federal court decision in 1993 abolishing weighted voting in New York state. In 1996, after a protracted fight, the Nassau County Board ofSupervisors became a 19-member legislature, each member having just one voteand representing districts of roughly equal population.

9. The United Nations Security Council The United Nations Security Council is aninternational body responsible for maintaining world peace andsecurity. As currently constituted, the SecurityCouncil consists of 15 voting nations – 5 of them arethe permanentmembers (Britain, China, France,Russia, and the United States); the other ten nationsare nonpermanent members appointed for a two-yearperiod on a rotating basis.

10. The United Nations Security Council To pass a motion in the Security Council requiresa yes vote from each of the five permanent members(in effect giving each permanent member vetopower) plus at least 4 additional yes votes from the10 nonpermanent members. In other words, a winning coalition in the SecurityCouncil must include all 5 of the permanent members plus 4 or more nonpermanent members.

11. The United Nations Security Council All in all, there are 848 possible winning coalitions in the Security Council–too many to list one by one. However, it’s not too difficult to figure out the critical player story: In the winning coalitions with 9 players (5 permanent membersplus 4 nonpermanent members) every member of the coalition is a critical player;in all the other winning coalitions (10 or more players) only the permanent members are critical players.

12. The United Nations Security Council Using a few simple (but not elementary) calculations,one can find that of the 848 winning coalitions, there are 210 with 9 members andthe rest have 10 or more members. Carefully piecing together this informationleads to the following surprising conclusion: the Banzhaf power index of eachpermanent member is 848/5080 (roughly 16.7%), while the Banzhaf power indexof each nonpermanent member is 84/5080 (roughly 1.65%).

13. The United Nations Security Council Note the discrepancy in power between the permanent and non-permanent members – a permanent member has more than 10 times as muchpower as a nonpermanent member. Was this really the intent of the UnitedNations charter or the result of a lack of understanding of the mathematics behind weighted voting?

14. The European Union The European Union (EU) is a political and an economicconfederation of European nations, a sort of United States of Europe. As of thewriting of this edition, the EU consists of 27 member nations (Table 2-8), withthree more countries (Turkey, Croatia, and Macedonia) expected to join the EUin the near future.

15. The European Union

16. The European Union The legislative body for the EU (called the EU Council of Ministers) operates as a weighted voting system where the different member nations haveweights that are roughly proportional to their respective populations (but withsome tweaks that favor the small countries). The second column shows each nation’s weight in the Council of Ministers. The total number of votesis V = 345, with the quota set atq = 255 votes.

17. The European Union The third column gives the relative weights (weight/345) expressed as a percent. The last column shows the Banzhaf power index of each member nation, also expressedas a percent.We can see from the last two columns that there is a very closematch between Banzhaf power and weights (when we express the weights as apercentage of the total number of votes).

18. The European Union This is an indication that, unlike in theNassau County Board of Supervisors, in the EU votes and power go hand in handand that this weighted voting system works pretty much the way it was intendedto work.